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• Alexander Grothendieck, Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas (Troisième partie) Inst. Hautes Études Sci. Publ. Math. 28 (1966), 5–255. (numdam:PMIHES_1966__28__5_0/)

motivation – local systems in physics:

Hilbert spaces of gapped quantum ground states over classical parameter space $P$ of topological phases of matter

topological phases are $C \in \pi_0(P)$

topological orders in phase $C$ are $\mathcal{H} \in Rep_{\mathbb{C}}\big(\pi_1(P,C)\big)$

topological orders in any phase are $\mathcal{H} \in Loc_{\mathbb{C}}(P)$

in fragile crystalline phases: parameter space is crystal lattice couplings reflected in 1-electron Hamiltonians from a space $\mathcal{A}$, hence parameter space is $P = Map(T^d, \mathcal{A})$

topological phases are $C \in \pi_0 \big( Map(T^d, \mathcal{A})\big)$

topological orders in phase $C$ are $\mathscr{H} \in Rep_{\mathcal{C}}\Big( \pi_1 \big( Map(T^d,\mathcal{A}), C\big) \Big)$

topological orders in any phase are $\mathscr{H} \in Loc_{\mathbb{C}}\big(Map(T^d, \mathcal{A})\big)$

functorial at least in diffeos of $T^d$ (modular functor)

$\mathscr{H} \in Loc_{\mathbb{C}}\big(Map(T^d, \mathcal{A})\sslash Diff(T^d) \big)$

for crystalline structure use equivariant maps

translate to TFT language

$$\mathbb{1} \longrightarrow Loc_{\mathbb{C}}\big( Map(-, \mathcal{A}) \sslash Diff(T^d) \big) \;\colon\; B Diff(T^d) \longrightarrow Vec Mod$$

in experiment, such FQ(A)H systems are governed by two effective symmetries

• supersymmetry

• area-preserving diffeomorphisms ($W_\infty$)

same as characteristic symmetries of super $p$-branes

$\rightsquigarrow$ find geometric engineering on M-branes in SuGra

$\rightsquigarrow$ need global IR-completion where topological brane charges are determined

(no string folklore, but actual definitions and proofs)

compare IR completion of Maxwell: $B^2 \mathbb{Z} \times B^2 \mathbb{R}$ and use Deligne complex…

…

magnetized M5 @ A1

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Theorem

(Milne 2017, Propositions 3.24, 3.25). Let $k$ be an algebraically closed field. The categories of affine varieties and affine $k$-algebras are anti-equivalent.

The following result particularizes the fundamental theorem on morphisms of schemes to prevarieties.

Proposition

Milne 2017, Propositions 5.11. Let $k$ be an algebraically closed field. Let $V$ be an algebraic $k$-prevariety. Let $A$ be an affine $k$-algebra. Then we have the following natural bijection:

$$Hom(V,Spm(A))\cong Hom_{k\text{-algebra}}(A,\Gamma(V,\mathcal{O}_V)).$$

In other words, the maximal spectrum functor and the global sections functor, defined between the categories of affine $k$-algebras and $k$-prevarieties, are mutually right adjoint. Note that Milne states the result for quasi-compact varieties, but his proof applies in the general case and never uses quasi-compactness nor separation. Note that from we recover .

For additional information, look at this very related typing graph.

$$\frac{\Gamma \vdash A ,; \Gamma \vdash B}{\Gamma \vdash A\times B}$$